Just finished watching the replay. I don't want to sound like an ass, but I don't think the KV-1s was your biggest problem, you made quite some mistakes in the early game. Some general things I noticed in the replay:
You spent a lot of time wiring cover off with your second pioneer, and I'm wondering if that was worth it. You could've gotten a grenadier instead of that pioneer, and cap stuff. Your opponent had better map control for pretty much the entire game.
Your MG42 and paks were often wandering unsupported, forcing you to retreat the MG42, or lose the pak. Also, you kept your paks too close to eachother, which made it pretty easy to flank them. If you keep them at a distance from eachother you can let them cover eachother.
I didn't really understand why you upgraded your PGs with schreks. At the moment you did that there was no vehicle threat at all, and it makes them a lot less viable against infantry. You needed anti infantry power, not AT power, so it was kind of an odd choice. If you didn't upgrade those PGs you could have used them to defend your Paks/MGs against those charging Shock Troops. Those 120 munis could have then be spent on Teller mines, which could've helped you a LOT.
You should've been a little more careful with your vehicles. You often just charged infantry with the SC, causing a lot of damage on it. There's no reason to do that, it does plenty of damage at range. You lost the Panzer IV because you repaired it at the frontline rather than in your base. The Panzer IV/Pak combo really paid off for you, so it was a shame to see that thing go down. You were really unlucky with your Tiger though
Also, I'm not sure if saving up for the Tiger was the best thing to do. It allowed your opponent to dominate you while you were waiting for 100 fuel, something which would've been harder had you gotten a PIV or a Stug. The PIV seemed to pay off for you, so why not get a second one?
Edit: Forgot to mention that your medic bunker came way too late. When you've got a decent amount of infantry on the field getting one should be priority! You really need them to be able to take a few shots before being forced to retreat again, and to avoid bleeding too much manpower.